© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n110 K11n112

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,17,8,16 X9,19,10,18 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6

Gauss Code:

{1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 4, -9, 5, -10, 7, -11, 8}

DT (Dowker-Thistlethwaite) Code:

4 10 -14 -16 -18 2 -20 -22 -8 -12 -6

Alexander Polynomial:

- t-3 + t-2 + 3t-1 - 5 + 3t + t2 - t3

Conway Polynomial:

1 - 2z2 - 5z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{7, 2}

Jones Polynomial:

q-2 - q-1 + 1 - q2 + 2q3 - 2q4 + 2q5 - 2q6 + q7

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + q2 - q4 - q6 - q10 + q12 + q16 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 + 2a-4 + 2a-4z2 - 4a-2 - 9a-2z2 - 6a-2z4 - a-2z6 + 3 + 4z2 + z4

Kauffman Polynomial:

- 2a-8z2 + a-8z4 + a-7z - 5a-7z3 + 2a-7z5 - 2a-6z4 + a-6z6 + a-5z - 2a-5z3 + a-5z5 + 2a-4 - 8a-4z2 + 14a-4z4 - 7a-4z6 + a-4z8 + a-3z - 6a-3z3 + 13a-3z5 - 7a-3z7 + a-3z9 + 4a-2 - 22a-2z2 + 32a-2z4 - 15a-2z6 + 2a-2z8 + a-1z - 9a-1z3 + 14a-1z5 - 7a-1z7 + a-1z9 + 3 - 12z2 + 15z4 - 7z6 + z8

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, -2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                     1 j = 13                   1   j = 11                 1 1   j = 9             1 2 1     j = 7             1 1       j = 5         1 2 2         j = 3       1 1 1           j = 1       1 2             j = -1   1 1                 j = -3                       j = -5 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 111]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 111]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 17, 8, 16], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[13, 21, 14, 20], X[15, 22, 16, 1], > X[17, 9, 18, 8], X[19, 13, 20, 12], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 111]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 4, -9, 5, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 111]]
Out[5]=	DTCode[4, 10, -14, -16, -18, 2, -20, -22, -8, -12, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 111]][t]
Out[6]=	-3 -2 3 2 3 -5 - t + t + - + 3 t + t - t t
In[7]:=	Conway[Knot[11, NonAlternating, 111]][z]
Out[7]=	2 4 6 1 - 2 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 111]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 111]], KnotSignature[Knot[11, NonAlternating, 111]]}
Out[9]=	{7, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 111]][q]
Out[10]=	-2 1 2 3 4 5 6 7 1 + q - - - q + 2 q - 2 q + 2 q - 2 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 111]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 111]][q]
Out[12]=	-6 -4 -2 2 4 6 10 12 16 20 22 q + q + q + q - q - q - q + q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 111]][a, z]
Out[13]=	2 2 2 4 6 2 4 2 z 2 z 9 z 4 6 z z 3 + -- - -- + 4 z + -- + ---- - ---- + z - ---- - -- 4 2 6 4 2 2 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 111]][a, z]
Out[14]=	2 2 2 3 3 2 4 z z z z 2 2 z 8 z 22 z 5 z 2 z 3 + -- + -- + -- + -- + -- + - - 12 z - ---- - ---- - ----- - ---- - ---- - 4 2 7 5 3 a 8 4 2 7 5 a a a a a a a a a a 3 3 4 4 4 4 5 5 5 6 z 9 z 4 z 2 z 14 z 32 z 2 z z 13 z > ---- - ---- + 15 z + -- - ---- + ----- + ----- + ---- + -- + ----- + 3 a 8 6 4 2 7 5 3 a a a a a a a a 5 6 6 6 7 7 8 8 9 9 14 z 6 z 7 z 15 z 7 z 7 z 8 z 2 z z z > ----- - 7 z + -- - ---- - ----- - ---- - ---- + z + -- + ---- + -- + -- a 6 4 2 3 a 4 2 3 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 111]], Vassiliev[3][Knot[11, NonAlternating, 111]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 111]][q, t]
Out[16]=	3 3 5 1 1 1 q q 3 5 5 2 2 q + q + q + ----- + ---- + ---- + - + -- + q t + 2 q t + 2 q t + 5 4 3 2 t t q t q t q t 7 2 9 2 7 3 9 3 9 4 11 4 11 5 13 5 15 6 > q t + q t + q t + 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n111

K11n110 K11n112